How do you solve the triangle ABC given a=7, B=60, c=9?

Jan 10, 2017

Use the Law of Cosines to find the length of side b.
Use the Law of Sines to find A.
The sum is 180 for C.

Explanation:

Use the Law of Cosines to find the length of side b.

${b}^{2} = {a}^{2} + {c}^{2} - 2 \left(a\right) \left(c\right) \cos \left(B\right)$

Substitute, 7 for a, 9 for c, and 60 for B:

${b}^{2} = {7}^{2} + {9}^{2} - 2 \left(7\right) \left(9\right) \cos \left(60\right)$

${b}^{2} = 67$

$b = \sqrt{67}$

Use the Law of Sines to find A.

$S \in \frac{A}{a} = \sin \frac{B}{b}$

$A = {\sin}^{-} 1 \left(\frac{a}{b} \sin \left(B\right)\right)$

Substitute, 7 for a, $\sqrt{67}$ for b, and 60 for B:

$A = {\sin}^{-} 1 \left(\frac{7}{\sqrt{67}} \sin \left(60\right)\right)$

$A \approx {48}^{\circ}$

The sum of the angles is equal to ${180}^{\circ}$:

${180}^{\circ} = A + B + C$

Solve for C:

$C = {180}^{\circ} - 48 - 60$

$C \approx {72}^{\circ}$